10

;-—-~-.

LA-5743-MSInformal Report

I

scientia

C-9f--a

UC-34Reporting Date: September 1974Issued: October 1974

C(2PY

Hydrodynamics and Burn of

Optimally Imploded DT Spheres

by

~of the University of California

<1 LOS ALAMOS, NEW MEXICO 87544-i’

4 \’

R. J. MasonR. L. Morse

‘)/a Iamosc Laboratory

UNITED STATES

ATOMIC ENERGY COMMISSION

CONTRACT W-74 OS-ENG. 36

,

In the interest of prompt distribution, this LAMS report was notedited by the Technical Information staff.

PrintedintheUaikdStatesofAmericm.AvailablefromNationalTechnicalInformationService

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.

D

HYDRODYNAMICS

We review

AND BURN OF OPTIMALLY IMPLODED

by

R. J. Nason and R. L. Morse

ABSTRACT

the phenomenologvof optimized

DT SPHERES

laser-driven-.DT sphere implosions leading to efficient thermonuclearbum.The optimal laser depositionprofile for spheres is heuristi-

_l tally derived. The performanceof a 7.5-pg sphere exposed to

—1. its optimal 5.3-kJ pulse is scrutinizedin detail. The timing

==*’requirementsfor efficient central ignition of propagatingburn

~~m in the sphere are carefully explored. We discuss the difficul-~~ m$===- ties stemming from hyperthermalelectron production and thermalo~ l-’ flux limitation. We give the optimal pulse parameters form—s;=%, : spheres with maases ranging from 40 ng to 250 pg, requiring

s===from 50 .2to 150 kJ of Input energy, and the corresponding

F=~s=

optimal performancelevels. The dependenceof pellet compres-0

g — ~.- ‘. sion, heating and burn performanceon the pulse energy, time====a)

$=:scale, exponentialrise rate, peak power and intensLty,wave-length, and on the degree of flux limitationare all systema-

.~o— tically described.-’~m: __

.L. ___

I. INTRODUCTION

High compression,as a means of enhancing the

rate of energy releaae in syetems undergoingnuclear

reaction,was appreciatedat Los Alamos as early as

1943.1 Recently, Nuckolls et al.2 at Livermore have

announced a laser-driven,ablative implosionscheme

for the compressionof DT pellets, and Clarke et al.3

at Los Alamos have ‘summarizedthe results of calcula-

tions which predict that DT spheres and shells can

be laser-implodedto conditions such that net break-

even thermonuclearburn ensues. Also, Brueckner4 at

KMS Industrieshas discussed the implosionand burn

of DT targets. The near-term prospects for laser

fusion have been outlined in survey papers by Boyer,5

and by Nuckolls et al.6

More recently, Fraley et al.7 have described in

some detail the burn physics and expansive hydro-

dynamics,which follow from high compressionof DT.

The preaent paper complementsthis burn study and

expands the Ref. 3 letter with reaulta from an ex-

tensive investigationof the laser pulse shape re-

quirements,implosive hydrodynamics,pellet condi-

tions, and burn performancepredicted for the op-

timized implosion of spheres.

Section 11 reviews the phenomenologyof opti-

mized implosions. In Sec. 111 we give the results

of numerical studies which vary the optimal pulse

parameters and pellet mass.

Conclusionsare drawn in Sec. IV. Our calcu-

lations were done with the Ref. 7 computer code,

modified as described in the Appendix.

II. PHENOMSNOLOGY

Under laser irradiation,electrons near the out-

side of the pellet, i.e., neighboring the critical

surface, are heated by inverse-bremsstrahlungand

various anomalous mechanisms.8 In the limit of mod-

est laser peak power this heat ia transported in-

wards by classical (Spitzer9)thermal conductivity.

With sufficientlylow laser input power the thermal

wave is subsonic. Heat is transferred to the ions

by classical electron-ion collisions. From pres-

sure increments in the thermal wave and from the

1

reaction force to the expanaion of ablating ions be-

hind it, shocks are launched toward the pellet cen-

ter. The thermalwave front acts like a “leaky”

piston. It abuta an ablation surface, bounding

shock-compressedand shock-heated plasma; Beyond

this surface the density and pressure drop, and the

ions are expanding.

For an optimal implosion2$3the early power

level should be kept low, so that the first shock

launched is weak. Thereafter, the laser powershould be time-tailoredto keep the subsequentcom-

pression of the core adiabatic to a maximal degree.

This ia accomplishedwhen the rising laser intensity

continuouslygeneratesweak, overtaking shocks,

which firat coaleace to a strong shock just before

the center. Upon the collapae of this final shock

a high ion temperatureis produced, initiatinga

spherical thermonuclearburn wave, which propagatea

out through the core of the pellet, consuming it.

A. Optimal Pulse

For a heuristic derivation of the optimal pulse

ahape consider that the pulse is driving an element

of the pellet core towards the center at a apeeddR

v=-—.dt

Similarly,a second lower surface iadS

moving in at w = - —.dt

The two surfaces demark a

shell of mass Am, width AR = R(t) - S(t), and den-

sity p = Am/(4nR2AR). Let s = so, R = R0’ and AR =

ARO at t = O.

The optimal pulse brings both surfaces and all

the shocks launched from the ablation surface of

the core to the origin at t = T. The fluid in the

shell moves with a apeed u ~ v m w. The ablation

surface launches weak shocks, which produce adia-

batic fluid changes, ao u . c, the speed of sound%and thus u - T , the mean fluid temperature. Also,

T - ~Y-1, so v . w . p& with Y the effective

ratio of specific heata.-3

Then the assumption AR . R leads to p . R or-a

v --w --R , a = 3(y-1)/2.

dR/dt = ‘V = CIR* for the

at t = T, obtaining

1

R = R.(1 - t/T)m

Thus, we can solve

boundary condition R = O

, (la)

so c1= Roa+l/[(l +a)T1. Similarly,

1

s - SO(L - t/T)m

, so AR = (ARo/Ro)R- R, proving

the assumption,i.e., p = *RO /ARo). It fOllOWS

that

,

&V = Vo(l - t/T) , V = Ro/[(l + Ci)7].(lb)

o

Finally, becauae work is done on the outer sur-

face at a rate W = 4nR2Pv with P - pT - R-3V2, i.e.,

W ? v3/R, and since we expect this to be propor-

tional to the energy input rate from the laaer, we

conclude that the optimized laaer exposure profile

is

IEo(l - t/~)-p, E ~E(tl)

i(t) =

0, E > E(tl),

(lC)

where p = (3a + 1)/(1+ a) = (9Y - 7)/(3y - 1)(=2

for y = 5/3), ;O is some appropriate initisl power,

and E(tl) is the chosen total laaer input energy at

shutdown time, t = t .I

This energy is given by

ioT—(l-Eq -(p_l) tl/_c)-(p-l), asauming

(2)

AtI = (T - tl)/~ <C 1.

The peak power input at shutdown is

-L L

( )&(tl) = to(LPl)p-l E(tl)/ioT ‘-1

-1 1

= (p-l)p-l(E(tl)p/ioT

)

px.

.

b

(3)9

2

‘0’5r—————710’4

10’3I {

P= I.65=.5.3kJ P=I.875-.

~=z.zo~:

10‘2~

‘l~472J

I162J f

Ii /0 10

.IL162J /;’

44J .H/./ r=3.125

,.10 23J ~P=L65 12J - /

:. L-0 Tz6.25—

0“.~ T=12.5—~-<p= 220

10$’- 5.2J

i-l

,0” ~o 2 4 6 8 10 12

Fig. 1.

t(ns)

Pulse shapes for the optimal implosionofa 7.5-pg DT sphere with T = 12.5 nsec andE(tT) = 5.3 kJ. D = 1.875.~. _ p = 1.65, snd—--p = 2.2.

X,t! , , \4LK5

.- .-Rkm)

Fig. 2. Optimized implosion.sequencefor the 7.5-pgsphere, p = 1.875, E = 2.15 X 109W. Firsteight frames for 6 +0= and 5.3 kJ of inputenergy; final four frames for f3= 1 andE(t.) = 7.5 kJ— density (g/cm3),---Te(keV), ii-tTi(keV), and ——<PR>(g/cmz).

Our calculationspredict thst high compression

can be achieved for p values ranging between 1.65

and 2.2. The correspondingeffective y run from

1.32 to 2.0. Figure 1 shows the pulse shapes re-

quired for the optimal implosion of 7.5-pg DT sphere

with T = 12.5 nsec and E(tl) = 5.3 kJ. The curves

are for p = 1.65, 1.875”,and 2.2 with ~. chosen in

each case to give maximum yield. Target perform-

ance details are given in Sec. III. Here it is im-

portant to note that the optimal pulse ehapes are

nearly identicalat the end, when most of the energy

is going in. In the last 500 psec, 90% of the

energy is delivered and the power rises by two

orders of magnitude. The p = 2.2 profile has the

lowest peak power -- 100 J/psec.

B. An Optimized Implosion

Figure 2 showa the optimized implosion dynamics

of a 7.5-pg aphere. The first eight-frame sequence

is for classical heat conduction,and results from

the Fig. 1 pulse with p = 1.875 and ~ = 2.15 Xo

109 “. The laat four frames assume a thermal epeed

limit on the electronicheat conduction,and re-

quired pulse retuning to ~. = 2.65 x 1C19 W, The

yield ratio (energy produced/energyinvested) YR =

Y/E(tl) is calculated including the effects of cl-

particle and neutron recapture during the thermo-

nuclear burn. The dashed curves are the integralR

ofpdR of the density of solid DT, and the

edge is at R.~=203 p, so the full integral

> pdR= 4.3X 10-3. The wavelength is

oBy t = 9.”9in the implosion sequence,

at the edge are at 450 eV and more than 102

pellet

<pR> E

10.6 p.

electrons

timea

hotter than the surface ions. A firat shock has

been launched toward the center and there haa been

some shock-overtaking,ao that peak density behind

the shocks is 2 glcm3 for a net tenfold compression.

By t = 12..44nsec the leading edge of the overtaking

chock envelope is down to 19 U, while the critical

surface (where p = pc = 4 x 10-5 g/cm3) haa moved

out to 0.12 cm. The laser ahuts down at t = 12.466

when the central denaity P(0) is 1.8 g/cm3 and when

collapse of the earliest shocks haa raised T,(O)

above Te(0) to 1.7 keV.

Burn has commenced by

yield ratio is YR = 0.014.

8.3 and bootstrap-heating,

J.

t = 12.4872 when the

At the center T1(0) =

chiefly from a-particle

3

.. .redeposition,haa raised Te(0) to 8.6 keV. The cen-

tral density has been compressedto P(O) = 2.3 x 103

g/cm3 and <oR> = 2.02. The ion temperatureexceeds

3 keV out to R = 1.57 p. This radiue defines an

initial central hot apot that includes 58 ng of DT.

The hot spot ia still collapsingon the origin at a

mean speed of 40 p/naec; its mean.temperature<Th>

= 4.8keV. ,Byt -12.4880 burn during this final

implosionphaae haa rai8ed <Th> to 8.9 k.eV,YR =

0.067, the hot apot haB etarted to expend, and <pR>

is at its msximum value, 2.09.

By t = 12.491 a propagating sphericalburn wave

has rataed Ti above 8.9 keV out to R = 14 p, which

includes 3 pg of DT, and 90% of the yield has been

released. All the yield, 177 kJ, is out by 12.56

nsec, when Y = 33.6, and a blaat wave in the ex-R

panding plasma ia heating the ions near R = 600 V.

The final electron temperatureprofile is relatively

flat due to the high thermal conductivityy derived

from the high T eatabliahedby both the burn and the

terminal laaer deposition.

c. Burn Conditiona

High compreaaion improves the yield from laaer-

heated plasmaa by decreasing ratio of burn time to

expansion time, and by raiaing the probabilityof

u-particle and neutron recapture. At high compres-

sion the best initial temperaturefor the fuel in

the pellet core is . 7 keV, since a-particle recap-

ture then raises Ti into the optimal 20-keV range

prior to the expansion of the core, generatinggood

yield for a minimum of energy invested. Similarly,

central deposition of the energy is desirable, since

then propagatingburn can heat the remainder of the

core. Details were given in Ref. 7; here we susma-

rize its nomenclatureand results.

For expansion times that are short compared to

the characteristicburn time, Te << Tr, the frac-

tional burn-up of DT microsphere is given by

f = Te12Tro r

(4)

with Te = R/4Cs, R the microsphere radius, Ca(Ti,Te)

(+

<Uv 1the speed of sound, and Tr = , in which <uv>

‘iis Tuck’sl” averaged cross section and mi is the

‘!DTion” masa (=2.5 amu).

Thus

and

f()

<Uv>

ro ‘~PR .

(5a)

(5b)

The parentheticterm in (5b) has a broad maximum

(= 1/11) over the range T = Ti = T = 20 + 70 keV,e

so

fr. = PR/1.1, pR<< 1. (SC)

When there is considerableburnup, fro ~ 0.1, de-

pletion of the fuel slows the energy production

rate, and (SC) goes over to

f- PR

ro 6.3+PR ‘pR S 1.0. (5d)

With full burnup 326 kJ are released per microgram

of equimolar DT, so the yield from uniform micro-

sphere is

Y. m

I = 326 f (kJ/lU3)0 ro”

Neutrona and a-particleshave mean free paths

WR‘ /(l+12:;:e5/4) ‘R

. 0.241PR (for Te= 10 kev) and

‘+ d 4.’’PR.

(5e)

(6a)

(6b)

.

I

.

.

4

t

.

In our optimally imploded spheree, as the bum

commences YR= 0.01 + 0.1, the density is either

flat at p(0) or gradually rising in the pellet core

up to 1.3 + 2.0 p(0) at the ablation surface. Be-

yond, it, in the blow-off, p rapidly declines as

- R-3. Thus, for pellets it is good approximation

to substitute<pR> for pR and to uae the core radius

and mass for R and m. in Eqs. (5) and (6).

For <pR> > 0.24 (Te < 10 keV), Eq. (6a) shows

that a-particlesare recaptured in the core. When

T > 4 keV energy production from burn exceeds the

pure bremsstrahlungloss. Consequently,the fuel

will bootstrap heat to a higher burn temperature.

At R = 2 and T = 6 keV, for example, Ref. 7 Fig.

(lOa) shows that we get a yield correspondingto

T = 15 keV, i.e., a tenfold improvementin the

output from bootstrap-heating.

The specific internal energy of equimolarDT

at degenerate densititea is

10 = 5.8 X 10-2ITi(keV)

t

+ Tef(keV) [1 +~(+)i ..1} kJ/p;7)

The Ref. 7 Go and Mc predictions for DT micro-

sphere neglected the effects of neutron recapture.

From (6b) we see that recapture should enhance boot-

strap-heatingand propagation for <PR> ~ 4. Con-

sequently,we introduce the multiplier,

(8b)

to measure the yield increase Y. + Y;, from neutron

redepositionIn uniformly heated microapheres and

‘we define

~ = (Yn/Y;)/ (l/l.) (8c)

aa the central ignition multiplier,when both a-

particle and neutron recapture are significant.

Laaer energy E(tl) couples through the implo-

sion process to the core of a pellet with an

efficiency,

E ~ moIc/E(tl) (8d)

with Tef = 5.7x 10-’ 1#3.

The bum performanceof uniform microapheres

is measured by the gain factor

YGo=-&- .

00(8a)

When there is a central hot spot from the final

shock collapse in optimized implosions, the addi-

tional multiplier, M = (Y/Yo)/(l/Io),measuresc

the benefits of propagating burn. The Mc multi-

plier includes the effects of (a) decreased Y from

the finite tranait time of the burn wave across the

core, and (b) the decreased 1 requiredwhen a hot

spot initiates the burn. Reference 7 showed that

for <pR> > 1.0, GO(T = 6 keV) > Go (T = 20 keV)

for at leaat a 3.3-fold decreaee in the input

energy requirementsto a uniform microsphere. Also,

it gave the rule of thumb that NC > 1 for PR > 2.

in which the core energy moIc is partly established

by bum-preheat during the core compression to max-

imum ~pR~. Also, the internal energy of the core of

a pellet may differ from that of its corresponding

microsphere (having the same <pR> and mean T values),

because of the density dependence of internal energy

that comes with degeneracy. A final multiplier

MI = 1/1c

(8e)

corrects this discrepancy.

Thus, our measure of overall pellet performance

becomes

Y EY‘R ‘zt~=~’

c Go@nMI.(9)

10’I ~

10s .

.—. ——— ——‘!=12.4892>‘--*, \

102 .1X~2~4%-4Z- -*,

10‘

\

10°#\,

10-s 10-4 10-3 10-2R[cm)

Fig. 3(a). Propagatingburn in the 7.5-pg spherefollowing ignition. density--- temperature.

102 I I—---- 1 I I I 1

-—-— -_-7- —-—_:Go

10’

=___10’J

-——— _ _

ImoI lkJ)

10-’L ! I 1 1 I 1

5 6 7 8 9 10 II 12CTh>(keV)

Fig. 3(b). Temperature dependenceof the burn per-formance mu.ltipliera.

In the Fig. 2 implosion the maximum <pR> is

2.09 at t J=12.4880. Since bootstrap-heatingsub-

sequently raises the core above 20 keV, a burnup .

fraction, fro = 0.25, is implied by (5d). The total

yield of 177 kJ then tells us through (5e) that the

active core mass with uniform initial heating would

be 2.17 pg. However, the Fig. 2 burn starts from

a central hot spot, and Ref. 7 showed that under

optimal conditionsonly 70-75% of the uniform yield

is extractedvia propagatingbum, so we take m. =

2.17/0.75 - 2.89 pg. l%is constitutes 39% of the

total pellet mass. At peak <pR> we calculate moIc

= 310 J, so E = moIc/E(tl) = 0.059. This includes

100 J of recapturedenergy generated during the laat

picosecondof implosion. The inner 58 ng (~) of

the core is at an average temperature<Th> = 8.9

kev. The remainder of the core ia at an average

temperatureTc = 0.49 keV. Figure 3(a) detaila the

progress of the spherical burn wave that ignites the

cold region in 3 psec, crossing it at a apeed of

-h x 108 cm/sec. The temperaturedependence of

our various multipliers is plotted in Fig. 3(b).

Its results represent an extension of the Ref. 7

bum study to the present special case, where m. =

2.89 Pg, Tc = 0.49, <PR> = 2.09, and fh - ~lmo

= 0.02. Figure 3(b) shows that at <Th> = f?.9,G

. 72, ~= 6.8, Mn = 1.03, and since moI =0.35 ~,

%=1.13. Thus, the bum study results for the

Fig. 2 implosion parameters predict a yield ratio

YR=cGoM~MnM1 =

(0.059) 72 (6.8) 1.03 (1.13) = 33.6,

indicatingthe origins of the ratio computed in the

full implosioncalculation.

Figure 3(b) makes an important addition to

the earlier burn study, by showing that although

Go (PR = 2) f.mproveaas T = 10 + 3 central ignition

essentiallyfails for T c 6.5 keV.

Also, it should be noted that when the neutron

redepositionla ignored, so that the neutrona are

allowed to freely escape from the pellet, then the

optimal yield ratio dropa to Y = 25.8.R

Correspond-

ingly, Mn = 1.0 and ~+ Mc = 4.8,

.

v

*

.

D. Compressionand Shock Heating

Compressionof the core should be carried out

as adiabaticallyas possible to keep the energy de-

mands on the laser to a minimum. This means that

the internal energy will at times be near its dege-

neracy floor. For Te + Ti + O, (7) shows that I +

3.3 x 10-4 pz’s kJIVg, which is 69 J/Ug at P =

3 x 103 glcm3 , or the thermal equivalentof 0.59

keV at classical densities. Clearly, this influ-

ences the energy requirementsof the cold outer

region of the Fig. 2 core, prior to its heating by

propagatingburn. Another energy aink is the +.6

J/pg needed for full ionizationof the DT by the

early shocks in the implosion sequence. These deg-

eneracy and ionzation energy effects are fully inr

eluded in the equation-of-statetables accessed in

our calculations. Degeneracy should S2S0 affect

thermal transport to the core. Still, classical

Spitzer transportcoefficientsare used here. As11 the proper forma ‘orpointed out by Brysk et al.,

the degenerate coefficientsfor I_aaerfusion condi-

tions remain somewhat uncertain. From the use of

approximate, interpolateddegenerate coefficients

Brysk reports only minor changes in the timing re-

quirementsfor optimal implosions.

The first shock in our calculationsis usually

strong (its Mach number M >> 1) with the DT

started at 1W7 keV (l”K). If we take the 2.5-fold

compressionpoint (where P = 0.54 glcm3) to be the

location of the first shock, then in the Fig. 2

optimized implosion, for example, we find the shOck

to be at R = 1.95 x 10-2 cm, when t = 0.3 nsec,0

and moving at V. = Ro/2T = 7.8 x 105 cmlsec, in

agreement with (lb) for ct=l. Also the R(t) and

v(t) values for this point follow (la,b) to within

10%, for at least the first 10 nsec. Convergence

increases the shock speed, so that by t = 9, v =

1.3 x 106 cm/aec. At t = 12.455, just before the

collapse of the first shock at the origin, v = 2.4

x 107 cmfsec.

Although the first shock is strong, the temp-

erature establishedby its central collapse is reg-

ulated by V. and ultimately~o. One wants v. to

be great enough to avoid premature shock-overtaking

outside the orgin by the stronger shocks which

follow. But also, V. should be kept low to mini-

mize the heating from the central collapse of the

first shock itself, which would limit the compres-

sion achieved from the follow-on shocks. Both crit-

eria can be satisfied, if T is large enough. Our

simulationsshow that T 7 10 nsec suffices for pel-

let masses in the range 40 ng < m < 250 Ug.

One Interpretationis that in the optimal

scheme each shock falling on the origin cushions the

one after it. AS the central density Q(O) rises,

strong shocks driven by a given pressure P fall on%

the origin with a decreased speed, v = (4P/3p(0)) ,

relative to the speed of the same shocks without

precompression. With decreased v the final shock

Nsch number and central heating are reduced, leading

to a higher <pK>. Judicious timing preserves just

enough of the heating for an optimal initiation of

propagatingburn.

Figure 4(a) displays the major Pellet Proper-

ties in the Fig. 2 implosion at the instant of laser

shutdown, t = 12.466. Near the origin Ti > Te~

from the collapse of the earliest shocks. In the

region beyond 13 p the pressure drops rigorously as

R-3. Outside R = 102 u the density acquires this

same dependence,P-R-3, while the Te profile be-

comes flat, and Ti - R-2. The rapid Te decline

near R = 13 u marks the location of the ablation

surface. The flow velocity reversal, and the P and

p drop off beyond this surface also identify it as

a deflagrationfront.12-lqThe fluid _Jufitahead of

the front is moving at u = 2.9 x 107 cmfsec, and the

front itself ia falling on the origin at UD = 3.2 x

107 cm/aec, measured from the trajectory of the

density peak. Thus, the front is penetrating the

mass ahead at u.P=%-u = 3.o x 10I3 cmjsec. The

-11 ‘peed ‘atio’ ‘P’%= 0.094, is consistent

with the sharp density drop across the front. But,

since the maximum sound speed is Cfi= 6.6 x 106 cm/

sec (at R = 12 v), we conclude that the Chapman-

Jouguet condition,UD = u + Cfi, is y3J strictly

obeyed. This is not surprising, in view of the

highly convergent and time-dependentnature of the

problem at t + T. The peak density rises from 400

to 680 g/cm3 in just the 5 psec preceding shutdown.

The insert in 4(a) shows the effects of pulse

detuning on the density profile at laser shutdown.

The solid curve repeats the main ffgureta profile,

which is for the optimal initial power, ~ (= 2.15

x 109 w). With ~ = 3.0 x 109 W the first shock

hae already hit the origin, raising the density

prematurely. We shall see below that this provLdes

,O.ti164 10

102(k V)

0“1.

/(

\-N

1\\‘\\ 1.

I , t , , II , , 1 ., ‘

. 10’ ~ , 1 t , , ,1 1 , , , ,,, , , , ,,9

-1

\, ‘\vTix102(W) i

\ \ \-1

-\-%1

, \,!, ,)l \ , , 1

10-2 10-’R (cm)

Fig. 4(a). Conditions in the 7.5-pg sphere at themoment of laser shutdown, t

I= 12.466.

—- T x 102 (keV),—–—P X 1A-2 (MJ/cm3),— P(61cm3),- u x 10-7(cm/see). Insert showsthe density profiles at shutdown whenE = 2.15 x 109 W (which is the optimalt&ing correspondingto.the main figure)‘, and detuned t? E = 1.5 x 109 w——— and to E. ~ 3.0 x 109 w----------- .

too much of a cushion for the later shocks, so that

1.4 keV is the highest central temperatureachieved,

end ignition fails. With ~. reduced to 1.5 x 109 W

the profile manifests premature shock-overtaking,

which results in a 20-keV collapse temperatureand

inefficientpropagatingburn.

In Sec. A we diacuased the implosionof a maaa

layer within a pellet to motivate our choice for

the optimal pulse. Figure 4(b) tracks the evolution

of one such layer in our Fig. 2 implosion. The

layer waa chosen to be inside the ablation front

and at the mean density at the commencementof

burn. Initially, its position is R. = 90 p and its

width ARO = 0.61 p. We have msrked the total

energy deposited by the time the layer has reached

its specified R. The layers p, and T and u values,

and the pellets <pR> are plotted versus R(t).

H’,

p(9/cm3)I

1!-3-R

T,x102~103 (keV)1

>I 12.46Sns(5.26kJ)shutdown

I\ ..#\\

.102

\

\\12.435(3.03)

\\\\\ [2.371U.66kJ)

\

-1

10 ‘ -1-R

[,k\,,,,-

\

12.324(751J)-7Uxlo

(cm/s)y_% \\ 12.197(482)

\ \

\11461(245)

10° I\

I ‘(%m2)\yj10.082(110)

I \

I

I( 9.091ns[66J)\

10-’lci4 10-3 10-2 10+

R(cm)

Fig. 4(b). The evolution of conditions in a masslaye~, initiallyat R - 90 U, in the7.5-pg aphere, aa its”optimizedimplo-sion proceeds,‘p(g/cm3)------- Ti x 102 (keV),———u x 10-7 cmfaec, and — <pR> (g/cm2).

There is no motion of the layer until the

first shock strikes it at t = 9 nsec. There fol-

lows a strong shock compressionproducing p u 0.84

g/cm3 by t = 10 nsec. Then, as the energy deposi-

tion runs from 300 J to 3 kJ, we see that p _ R-3,

T- R-2, and u - R-l, as predicted in Sec. A for

p = 2 (Cl=l)--even though p = 1.875 is, in fact,

used here. Central shock heating and burn retard

the compression for R ~ 5 p. The kinetic energy of

the shell starts to go over to thermal energy, and

then burn takes the temperatureup beyond 15 keV.

Expansion of the central pellet hot spot, as prop-

agating burn begins, then pushes the shell density

over 7 x 103 g/cm3. Note that at laser shutdown

<pR> is already large enough (-0.25)

recapture in the pellet.

for a-particle

.

s

8

*

4

102 I I8111111 [ 1 I “1

10‘

10° .

Y2.. [—~ 10-’

10-2

10-3

IIIO-41 J , ,,,,,,1 I [ I J.-

10-’ 10° 10’ _ 102 103 104p (g/cm3)

Fig. 4(c). The T , P adiabats for the optimizedFig. ~ implosion. Trajectoriesaregiven for the layer discussed in (b)~nd for the central calculationzone.Eo= 2.15 X 109 W

Finally, we return to the question of pulse

tuning. Figure 4(c) shows the significantTi, p

adiabats for the optimized Fig, 2 implosion. The

lower curve shows the evolution of the Fig. 4(b)

mass layer. The upper curve tracks the conditions

in the central zone of the pellet. The mass layer

is shock compressed to - 0.84 g/cm3, and 1 eV and

then follows a T - p2/3

adiabat until p > 2 x 103

glcm3. The central zone seea a stronger convergent

shock that compresses it - thirtyfold to 6 gfcm3

and onto a temperatureplateau near 4 keV. When

the yield starts to come out, YR=O.O1, the layer is

highly degenerateat p = 3.6 x 103 and only 0.25

keV. During the burn it is heated to over 35 keV.

Mcst of the yield is out by the time the layer and

center have expanded down to p = 150 glad. FOllOw-

ing laser shutdown and arrival of the firat shock at

the center, there is some, slight conductive cooling

of the center down

shocks coalesce at

compressionof the

to 2 keV. Then, aa the remaining

the origin, there is adiabatic

central zone to p = 2 x 103 g/cm3

--fYP=5.67. L

.1 I 1 I I WAl

10° 10’2

[03 104

p(g/cm3j0

Fig. 4(d). The adia~ats forand for E = 1.5correspondingtothe 4(a) insert.

i x 3.0 x 109 w —--,X“log w *’the detuned cases in

and 5 keV. Burn during the final implosion phase

raises the zone to 7.5 keV (YR = 0.01), and then

bootstrap-heatingtakes it up to 65 keV. This com-

plex behavior is concomitantwith optimal yield from

the 7.5-Bg pellet.

Figure 4(d) shows the response to mistuned

pulses. When llo= 3 x 109 W the central zone shock

heata to only 0.3 keV. The first shock is premature

ao there follows a subsequent expansion down to 30

eV. All this occurs well before laser shutdown--

unlike the tuned case where the arrival and shut-

down are nearly simultaneous. Following shutdown

the~o = 3 x 109 W pulse brings both the layer and

the center up to 1.5 x 103 g/cm3. But, since the

center goes only to 1.8 keV, there is no ignition

and negligible yield. On the other hand, with ~0

too low at 1.5 x 109 W, the first shock takes the

center directly to 15 keV at p = 0.84 g/cm3, The

subsequent compression is nearly isothermal,due

to the good thermal conduction above 15 keV. When

YR = 0.01 the center is at 20 keV, and still com-

pressing. The high central temperaturelimits the

9

maximum density achieved to 1.2 x 103 g/cm3 in both

the layer and the center, and becauae of the lower

<pm achieved (=1.20here, as compared to 2.09 with

the optimal pulse), there is only time to raise the

layer to 12 keV prior to expansion. This limits YR

to 5.64.

E. Thermal Transport and Flux Limitation

Electron thermal conduction transfers the laser

energy from the critical surface to the ablation

front. The electrons are highly collisionlessnear

the critical surface. Reference 15 carried out sim-

ulations of the collisionlessconductivity,and

showed that where the total electron density is n,

the minimum & hot electron velocity v needed toh

transportan energy flux q (W/cm2) is given by

= 1/16. (lOa)

This assumes planar, one-dimensionalgeometry,

steady transport,and that the minimum vhoccurs

with nearly flat hot and cold distributionfunctions,

such that vh= (3kTh/me)%,and Th = 2T relates the

etemperatureof the hot electrons to the mean temp-

erature. When (lOa) ia expressed in terms

and adjusted to include energy flow in three dimen-

sions, it becomes

‘.lIux . !3+(ye) (lOb)

()kT +with ;=;-# and f?= 1.0. This agrees with

ethe work of Forslund16, and the limiter employed

in the Livermore Lasnex2$6 simulation code. If the

laser-generateddistributionis stable but off-

optimum, (lOb) tells us that Th is above the mini-

mum and B c 1. Alternatively,the distribution

could be unstable,17with a long growth time from,

say, the presence of a very weak hot electron strea~

i.e.,~ln << 1, then S >1.

At the critical surface under limitation the3/2peak laser power obeys q(tl) - ncTc from (lOb).

Conaider a series of optimal implosions at diffe-

rent wavelengthsA. The peak intensity goes as

q(tl) --@/R~. In the blowoff n - R-s, the

critical density obeys nc ~ A-2, and simulationscan

provide r in the phenomenologicalrule fi(tl,A)- Ar.

Thus, the coronal temperaturewill scale as

[ dE(tl,A)Tc - (q/nc)2’3- UR

cc

2/3

(ha)

We shall see in Sec. 111 that, typically, r z -0.2,

so T - AO”3. l%is gives a 2.O-fold increase in

temp~raturefrom the cha~,geA = 1.06 + 10.6 p. sim-ilar3.y,as we go to different pellet masses at fixed

A, R -ml’3, and thusc

(z)i(tl,m)2/3

T2/3 _

‘c-q Rc

(llb)

~ m2/3(s - 2/3),

in which fi(tl,m)- ms can be derived from implosfon

calculations. In fact, we find that s = 0.36, so

Tc _ m-0”20s giving a twofold decrease in coronal

temperatureaa we go from, say, m = 7.5 + 250 pg.

At low-energy flux levels classical thermal

conductivityis accurate. To bring in limitation

smoothly at high levels,

(’e%)“we make the substitution

+ (=%’)-1+(12)

[%%$?)]‘1in our simulationcode [Eq. (b-2b) in Ref. 7].

.

.

.

10

The last four frames in Fig. 2 show details of

the optimized implosionof the 7.5-pg pellet for

B = 1. To maintain the optimal yield it has been.

necessary to retune the pulse to E = 2.65 X 109W,o

and to raiae the input energy to 7.5 kJ. Thus limi-

tation decreases the optimal yield ratio Y; to 23.6.

Also, the coronal temperatureat the critical sur-

face riaea from 12.3 keV with f3+ m to 48 keV when

,8=1. The Te profile minifeats a plateau near the

critical surface, but otherwise the implosionphen-

omenology is the same as for @ + ~. The greater

energy input s~ports the higher cor(onaltemperature.

The increase in E. maintaina proper timing of the

intermediateshocks, which tend to be delayed as

limitation slows the rate of rate energy transport

from the critical surface to the ablation front.

Use of the limiter can provide a first estimate

of the effects of hyperthermalelectron tranaport.

Generally, the results which follow show little dif-

ference between !3= 1 and f3+ m calculationsfor

A = 10.6 p, and this is encouraging. On the other

hand, flux-limiteddiffusion faila to model core

preheatia from any hyperthermalelectrons generated

in the corona, and misses geometriclgeffects as-

sociatedwith the angular momentum of electrons,

rattling about in the corona and possibly missing

the core. Furthermore,the effects of plasma insta-

bilities, from ion and magnetic field fluctuation

and from the time dependenceof the laser deposi-

tion itself, may lie outside the scope of any dif-

fusive tranaport treatment. For an accurate des-

criptionof these effects, a fully self-consistent,

kinetic transportmodel will be required. In the

interim, problems associatedwith hot electrons can

be reduced by going to shorter wavelengths andlor

to larger pelleta, taking advantage of the Eq. (11)

scaling rules for the coronal temperature.

III. PARAMETER STUDY RESULTS

A. io, E(tl), and ~(tl) Dependence

Figure 5(a) has been constructed from the re-

sults of many runs to show how target response

changes as ~. and E(tl) in the optimal pulse [lc)

are varied. The results are for the 7.5-~g sphere

exposed to 2.5, 5.3,and 12 kJ of C02 light with

P = 1.875, T = 12.5,snd 6 +-. The figure collects

much data, but it is instructiveto ahow it together.

First, we give the yield ratios observed for 12-

and 5.3-kJ input cases--theyield with 2.5 kJ ie

negligible. Then we show the maximum central den-

sities p(0) achievedwith the three input energiee,

and plot <PR> for the 5.3-kJ case--it obviously

tracks p(0). Finally, we include the central ion

temperaturesT(0); they prove to be an extremely

useful diagnostic in “tuning up” the pulse. For 12

and 5.3 kJ we give the T(0) registeredat the time

when 53 J has been released (thiscorrespondsto

YR = 0.01 at the.5.3-kJ energy--whichwe find to be

optimsl). When ~. exceeds 2.2 x 109 W the burn ef-

fectivelyceases, so here we give maximum T(0) ach-

ieved in the implosions. Similarly,we give the

maximum T(0) registered in each of the 2.5-kJ im-

plosions.

With 5.3 kJ of input, T(0) runa from 20 keV at●

E = 1.5 x 109 W to 1.2 keV at 3 x 109 W. The cen-0tral temperatureis 6.6 keV at the ~. of maximum

yield. The yield ratio rises slowly to its 33.6

maximum at 6.6 keV and then drops precipitously.

Observe that p(0) and <PR> have their maxima near

the optimal yield point. Pellet response to the

optimal tuning haa been carefully scrutinized in

Sec. II, B-D. Figure 3(b) shows the aharp decline

in the effectivenessof propagating burn below 6.6

keV. We conclude that the optimal tuning mskea

maximal use of propagatinrjburn.

When 12 kJ is supplied we get the same T(0)

dependenceexcept below 6.5 keV where the yield is

negligible. Generally, the p(0) values are higher

than in the 5.3-kJ runs. At the density maximum

T(0) ia 16 keV. There is some increaae in the ab-

aolute yield produced,but the input energy ia so

much larger that YR drops. Alternatively,with 2.5

kJ supplied T(0) is always below 3 keV for <pR~

above 0.65. The ~pR~ at maximum density is only

0.88, while T(0) there is 1.2 keV. There is no ig-

nition and negligible yield.

In general, when the input energy ia increaaed,

T(0) increases at the fiopoint giving maximum <pR>.

More refined calculationsplace the optimum E(tl) at

5.3 kJ ~ 10%. We conclude that for the overall

optimum tuning one must provide just enough energy

to put the maximum <PR> just above the 6.5-keV knee

in the central ignition curve, MC(T) of Fig. 3(b).

Thus, in optimized implosions~. controls the

central temperatureachieved, and compression is

linked to the energy supplied. The energy is

11

‘“’m 10‘

p x10-3( g/cm’lJ

10°

,L..-...I Ii

10-’ II I I I I I 1

0.5 -1.0 1.5 2.0 2.5 3.0 3.5 4.0

Eoxlo+(w)

Fig. 5(a). Pellet conditi~nsin the 7.5-pg spherefor different E. and E(tl) choices

.— YT(O) (keV),— ~?~-~~~~~~~nd~hh~ <pR> (g/cm2).

however, directly related to the peak power in op-

timal pulses [see (2)], so it is not immediately.

clear whether E(tl) or E(tl) is important to compres-

sion. Figure 5(b) demonstratesthat peak power is

the controllingparameter. It shows the performance

of a 0.7-pg sphere under 700 J of C02 light (1?+ CO)

with the energy deposited in accordancewith (lc)

(againT = 12.5 nsec, and p = 1.875) until a ceil-

ing rate ~c was reached. Thereafter, energy was.

supplied at the Ec rate until the full 700 J were

delivered, The figure plots the optimizedpellet.

conditionsvs E . With no ceiling imposed the peakc

power just prior to las~r shutdown is 85 J/psec

(8.5 X1013 W); YR= 13.2and ~o= 7 X107W. The

yield is off slightly at 45 J/ps, and drops rapidly

with further reductions in the ceiling. For break-

even ;c ~ 12 Jlpsec is required. Going from 45 J/psec

to 25 J/psec the optimal retuning in ~. compensates

for the lost <oR> by raising the peak central temp-

1‘‘‘“1 , , I 1 ,

1.-

flP(0) XIO-3( g/cm3)

v’”

/’

I/*/I

,“-1~5 10 30 100

kc ( J/Ps)

Fig. 5(b). Response of conditions in a 0.7-l/gsphere to a peak power ceiling, E=on the optimized pulse.

erature to 21 keV [which gives the best fractional

burnup (4) in the absence of bootstrap-heating].

With still lower ; these higher T(0) become lnacces-C

aible. With the ceiling imposed, no improvement in

performancederives from increases in E(tl). This

is not really surprising,in view of the arguments

preceding (l), which suggest that the density

achieved to a given time is proportional to the in-

stantaneouspower.

B. The Time Scale T

The results, thus far, have all

ed time scale T = 12.5 nsec. Figure

optimal performancechangea with T.

considering the 7.5-pg sphere with P

10.6 U, (~ + CO),and E(tl) = 5.3 kJ..

consideredE. was tuned to determine

been for the fix-

6 shows how the

Again, we are

- 1.875, A -

For each T

the maximum

‘R”This is plotted

pellet conditions.

along with the correapondin8

,

.

12

102

10 ‘

10°

10-’

I I I I I ~

1“,/- “—----,/

‘\\ / “\< T(0)l>-1

_J&~__e.4

i(tIlX10-3(J/PS)\ /.

.*

I I 1 I 14 8 16 20

7(:s1

(13b)

with ~V=~-tsnd1

ing point lies on an

the scaling rule

i’ = i (T/T’)p. The new start-0- 0extension of the old pulse, If

ioTp - const {UC)

is observed. The Fig. 6 results obey this rule, so

they all lie on the single optimal profile for 7.5

Ug, P = 1.875.

Only 23 J is deposited during the first 6 nsec

of the Fig. 1 p = 1.875 profile. One might choose

to start the deposition at t = 6 nsec to ease the1

laser timing requirements. Figure 6 shows, however,

.-J that the early deposition is quite important, since

Fig. 6. Optimal performancecharacteristicsfor the7.5-Ug sphere vs T.

The yield ratio has a broad

nsec. It drops rapidly ss we go

slowly as T approaches 20 nsec.

of T values, the product ~oTp is

maximum near 15

below 8 nsec, end

Over this range

very nearly con-

stant. This is significantbecause it tells us that

all the optimized profiles for different T are, in

fact, the same curve with different extensions to

early time.

To see this connection,consider the following.

Suppose we have found the optimal pulse for a given

T value, and we choose to deposit laser energy in

accordancewith this profile, but starting at a new

time tl < T. Then the new deposition rule is

(t +tl’ -p

E(t)=~o 1-—T )

(13a)

YR drops from 33.6 to 13, if T is reduced to ‘r=

6.5 nsec. The early part of the pulee is needed to

launch weak shocks to cushion the final shock col-

lapse. If T is too short, here < 8 neec, the early

shocks are themselves too etrong, generating exces-

sive preheat. Also, for T < 5 nsec & must be soo

intense to launch converging shocks that the ther-

mal front tends to burn through the ablation eur-

face it supports. Thus, with T = 3.1 in Fig. 6,

T(0) climbs to 18 keV. The improvement in p(0) and

<PR> as we go toward 12 nsec follows from a reduc-

tion of this preheat. On the other hand, the fall-

off in performanceas T + 20 nsec occurs because

enough time is svailable for the earliest shocks to

reflect and recede somewhat from the origin prior

to the arrival of the main, collapsing shock

envelope.

c. p, A and $ Dependence

Figure 7(a) determines bounds for our choice

of the exponent p. Again, the results are for the

7.5-pg pellet under the usual conditions. The best

YR is at p = 1.9. The dropOff is less than 10%

over the range 1.85 < p < 1.95, and less than 30%

for 1.68 < p < 2.1. There ia a rapid decline in

YR for p < 1.68, and a more gradual decline for

13

Fig. 7. Dependenceof the optimized performanceand pulse parameterson

’02~

I i

10‘

10°k/---/.—-{.,i

‘-i’\\ ‘\,; ‘\\

T(0) (kev)\

–l.____\_-‘ - . ---

i \\P(0)x10-3(g/cm3)

i \

/

~ jbb%

7‘*

!# <pR> -n(g/cmz] %+

“t

10-’ k I { I !1,6 1.7 1.8 1.9

I I2.0 2.1

P2.2

(a). The pulse exponent p,

102 - [ I I 1 I I I II I I I I I i I q

10’I(/3=1)

Tc [keV)

/“

/’____ T(0) ( keV),/ ——-—_ _~

-—— -_

/‘R

-1

{\ —---7---— ——--—---—< ~>

I0° (:/cqZ)I I I I 1 1 I 1I 1 I I 1 I 1!0.1 I.0

A (pm)10

(b). the wavelength A9

I 02 . 1 1 I I 1 J I 1 1 I 1 I\

I 1 [ In

\

‘y&fx)x IO-l( J/Ps)

\‘~_

‘_= —-- _@_U

-\- L

-- %.

10’-v

E(tl)(kJ)

‘-. ~ “’’)L;-—T -—-—-----

koxlo-g(w) [L?=l)

10° 1 t I t I t t 110.1

I I 1 1 1 tII

A(fm)10

(C). further dependents on wavelength k, and

10 ‘ /\’” t(?l)(kJ) -1

.- 0Iuv.\‘?’-’2“<pR>

(g/cm2)

Tcx10”2(kW)

lo-’~o I 2 3 4 5

I/#

.

(d). the degree of flux limitation1/6.

14

p > 1.9. The yield ratio passea through unity at p

= 1.62 and 2.3. The required initial intensityco

declines exponentiallywith p. At small p excessive

preheat ruins the performance;at large p the shock

timing ia such that a good <PR> is inaccessiblefor

T(0) > 8 keV.

Performanceaa a function of the wavelength is

described in Fig. 7(b,c). This is for the usual

7.5-pg sphere. We give YR, the coronal temperature

Tc, and the significantpulse parameters for both

~ +ca (no l~iter) and ~ = 1 [see (lr)b)].~,en i

> 10.6 p we anticipatehot electron problems,when

A ? 0.13 u the laser fully penetrates the uncompres-

sed DT target. We see that with 13= 1 the lsrgest

wavelengths give the best performance. As A de-

clines and the depositiongoes deeper, increased

E(tl) and fi(tl)are needed to produce a good <pR>,

and still <pR> at the optimal YR falls off. The

optimal i tuning for ~ + 1 is constant down to 1.060

p, and then decreases. The difference L? the yield

ratio between .6+ co and ~ = 1 disappears below 2

V; the difference in the coronal temperatureis

gone below 0.5 p.

Figure 7 (d)shows the optimized perfornmce of

the 7.5-pg pellet under C02 light as l/B = O + 4.

If the required value is ~ = 1, then the yield ratio

drops from 33.6 to 21.5; also the required optimal

energy rises from 5.3 to 7.5 kJ and Tc rises from

13 to 48 keV. IfEi- 0.25, YR iS down to 0.8.

Similar results have been obtained by Ashby and

Christiansonat Culham.20

D. Msss Dependence

A large nuuber of rums were made to determine

the optimal pulse parameter for (lc) as a function

of mass, and to examine the correspondingpellet

conditionsjust prior to burn. The calculations

were for spheres with messes ranging between 40 ng

and 250 pg, with the exponent p restricted to 1.875

and the wavelength fixed at A = 10.6. We conducted

a three-way optimization in ~o, T, and E(tl). For

each time scale we found the best ~. and E(tl) by

the proceduresdescribed in conjunctionwith Fig.

5(a). This was done over a range of T values (as

for Fig. 6) until the beat T was found. ‘Ibistun-

ing process was tedious and somewhat inaccurate,so

our T and E(tl) valuea are goOd tO about 2%. The

plotted results are principallyfor B + coalthough

the YR values for 6 = 1 are shown. There ia little”

reduction in the performanceat B = 1, especially

with the larger messes. This mass dependence data

is collected in Fig. 8 and Table I.Figure 8(a) shows that YR exceeds breakeven at

4S ng; Fig. ii(b)tells us this occurs with 60 J of

input energy. The yield ratio is 33.6 at 7.5 pg

and 65 at 250 pg with scaling YR -.m“”lg for m >

7.5-pg. The <PR> of the pellet exceeds 1 g/cm2 at

m = 0.1 pg. It goes above 2 at m = 2 pg and above

3 at 100 pg. Its scaling is <pR> -.mO”08 for m >

7.5 pg and deneity drops from 1.8 x 10* glcm3 at

70ng tol.03 x103 glcm3 at 250us. The central

ion temperatureat YR = 0.01, T(0), drops from 21

keV at 40 ng to 7.5 keV at 0.1 pg. For larger

masses T(0) is relatively constant.

The results at 7.5 Ug are essentially those

of the Fig. 2 optimized implosion except that the

pulse hae been retuned at T =.17.1 nsec with E. =

1.13 x 109 W and E(tl) is up to 6 Id. The <PR> of

the pellet is up from 2.09[Figs. 2 and 5(a)] to2.38.

But the optimized yield ratio is unchanged, YR =

33.6.

To maintain a fixed <PR> as we go to different

msases the density must change in accordance with

P(0) - <PR>312,ml/2 . (14)

Generally, it is desirable to have ~pR> S 2 for

efficient propagatingburn, or at least <PR> 5 1

for bootstrap-heating.

As we go to small masses, however, the density

must be made so large for <pR> S 1 or 2 that (a)

the a-particle mean free path becomes too long for

effective bootstrap-heating[ace Ref. 7, Fig. l(a)],

and (b) the large internal energy of degeneracy neu-

tralized the yield multiplication from propagating

burn. Thus, the 40-ng pellet ia optimized when

<PR> = 0.6 and p(0) = 1.6 x 104 g/cm3. Achievement

of <pR> = 2 would require further compression to

p(0) = 8 x 104 g/cm3. The optimized tuning in fio

sets T(0) at 21 keV, which is the best temperature

for burn in the absence of bootstrap-heatingand

propagation.

With larger masaes it is easier to get a good

<pR>, but excessive input energy may be required.

15

Fig. 8. Mass dependence of

.,.21 I I I -3

‘Rti-~<-”

:0 ;*: :

/A (g /cmz) (9/cm3)L J

10-’ 1 I I I

10-2 10-’0 10’ 10* 103

10 m(~g)

/

(a). optimized pellet performancecharacteristics,

102I 1 I I

YO/m (kJ/pg) ~

->---=:=

,.O-; -/--

Y/m ( kJ/pg)0’ . / ./”

‘,/

/;9

J ____~Eltx)/m [kJ/#g)100 . ------

Y/Y.

10-’ r

1,, ,,1 1 I

16*

I

10-’Ud

‘oom(pg) ‘0’ 10* 103

(b). further optimized pellet performancecharacteristics,

At 250 ug, <PR> = 3.1 with only p(0) = 1.03 x 103

g/cm3, but 150 kJ must be supplied. In any case,

the optimally imploded pellets manifeat only a grad-

ual <pR> increase with mass, since beyond <PR> = 2.5

fuel depletionseverelylimits any additional yield

from greater inertisl confinement.

As an estimate of the core mass we use

m =** (W3)o

(15)

with <pR> and

The resultant

It levels off

16

p(0) obtained from the Fig. S(a) data.

me/m ratio is plotted ~ Fig. 8(b).

at molm = 0.40 for m ~ 7.5 pg. Thus,

102~ I It it=)x16’

I 1‘1

[J/Ps)

10 ‘ 7

/“’” ( Wtpg )

100 /-/

./-

. ./- —-0

T-AQ--j~oA

10-’ I I I I UJ

10-2 10-’ d 10’ 102 103m(pg)

tt

(c). the optimal pulse parameters,and

m (~g)

(d). the half-intensityl/2%a

at peak power,the critical rad u~--------------

R at various wavelengths,and initial radiuaof the pellet edge—— —Re(t=()) .

the core constitutes 40% of the maas in larger, op-

timally imploded pellets.

The yield from uniformly heated cores is

mo<pR>

{

<pR> > 1Y. = 326 (6.3RPR>) ‘U)

(16)T(burn) > 20 keV,

baaed on (5d and e). We have plotted Ye/m in Fig.

8(b). Also we give Y/m = YRX E(tl)/m from the

full implosionsimulations,as well aa the ratio

Y/Yn. Clearly, Y/Y. levels off at 0.75. SO with

optimization,

extract - 75%

heated cores.

central ignition and propagatingburn

of the yield available from uniformly

*

,

Above 7.5 pg the optimized yield scales as Y

-ml.12. The requisite energy runs from 1.23 kJ/pg

at 40 ng, where the influenceof degeneracy is most

severe, down to 0.65 kJ/vg at 250 pg where prop-

agating burn is most effective. The energy scaling

iS E(tl) ~ mo.93a Together, these results scale

the yield ratio as YR= Y/E(tl) -.m“”lg.

Alternatively,YR . cGo~ from (9), assuming

Mn and ~ are constant. Above 7.5 pg at optimum,

10 is constant (at its - 7.5-keV value) and m/m. =

0.40, whileY/Yo = 0.75. Thus, Go g Y jm~ ~ ye/mo 00 0

-Y/m -m”.lz. Consequently,the product c< .

m0.07, and Mc Z (Y/Yo)/(l/Io)- l-l. Now, if we had

pure adiabatic compressionof the core, from con-

stant initial conditionsup to p(0), then I . p(0)2/3

-(m-.41)2/3-m.27 [from the Fig. 8(a) data]. Thus,

for a constant coupling coefficientE this would

say that c< --m“27, which is too large by a factorm.20. The discrepancy is explainedby the fact

that for optimal tuning the first shock crossing

the core must be progressive.lystronger,as the”

mass Increases. This is demonstratedbelow. Thus,

the initial core temperatureprior to adiabatic

compression ia correspondinglyhigher--raisingthe

requirementson I and E(tl).

Figure 8(c) shows that the optimal.time scale

runs from T= 8 nsec when m = 40 ng to T = 30 nsec

when m = 250 pg. Thus, for reasons still unclear

T . m116 -..i%~lz(t=o). Pellet performancedrops

rapidly if ‘cis too short, as evident from the Fig.

6 results. We find that a 12.5-nsec time scale is

sufficient for good performanceover the whole mass

range investigated,but at 250 pg, for example, YR

drops from 65 to 38 for a reduction T = 30 + 12.5

naec.

We find that the optimal initial pulse power

obeys the scaling rule

(17)

with q = 1.15. Our earlier Ref. 3 found that q =

1.5 with p = 2 and T fixed at 20 nsec. A rough

justificationof (17) follows from assuming that

the optimal initial power Iauches a first shock

obeying ~. - Pvs R2(t=O). The shock speed v~ and%fluid velocity V. vary as v -vo-(P/po) with PO

a

the density of solid DT - a constant, so P . v-z.

and E - v 3 ~z. With proper timing V. - R/T and

R .mY13, ~hile~-m116. Thus, finally, ~. - R5/

5’3h1’2-.m~”15,0

~3.m as observed. This empha-

sizes that the optimal tuning for all msssea gets

the first shock to the origin at t = T. Also, we

see that the first shocks are stronger for larger

msssea, being launched by an increasingpressure P.

-.POV02 - R~/T2 - m0”33.

The &o/~o curve In Fig. 8(c) gives the range

of ~. values about the optimum over which YR > 1

ia obtained for each mass. Clearly, the tuning

requirementsbecome less stringent at higher msss-

es, so that with a 250-pg sphere, more than 100%

deviation in ~. ia permissible. The permissible

error with spheres as targets is significantly

greater at low masses than with the shells dis-

cussed in Ref. 3. With a 2.7-pg sphere (absorbing

2.4 kJ) more than a 40% deviation in fiois allowed

as compared to a permissible 3% error in the Ref.

3 shell of the same mass.

The peak input power for the optimized implo-

sions varies as i(tl) - m0.34 as required by (3)1/6

when T - m . Thus, the final power runs from

29 J/psec at 40 ng to 550 J/psec when m = 250 yg.

Figure 8(d) plots half the peak intensity

required for the optimized C02 implosions at the

various messes, and gives the calculated C02

critical radii at peak power; 1/2 Qmax = ~(tl)/

8TR2. The half-intensityis recorded, since in ac

classical calculationof the thermal transport,

only half of terminal energy input flows from the

critical surface toward the core.

We see that at 250 pg the C02 critical sur-

face has expanded to 0.37 cm, so that the absorp-

tive surface area exceeds 1 cm2. We observe no

significant readjustmentin the location of Rc with

changes in T from 30 nsec to 12.5 naec and pulse

reoptimization.

The critical radii for 1.06 p and 0.17 p were

drived by noting that in optimally imploded pel--3lets the density falls as R , largely independent

of the depositionwavelength. Thus, we used,

for example,

TAB2.2r (b~

RC(l.06)

RC(10.6). ($::;; )1/3 .[@2/3 = ~ i& ~ k ) (J/pslc) 1 (10.6)u

,

,.

!2il!211(+3 E (1.06)11

373473101Xza34048o6.20

1.0

9.310.7

12.311.21.7.4.30.

2.71t20~— 299.3X306 .23 422.6X207 .36 M7.3X107.31 021.2X109.38 17sb.oxloy .47 2301.2311019.62 36$6.4XL010.93 560

167260320460low1570

0.04

0.11

0.27

0.10

7.5

22.

n.

250.

.03

.129

.292

.lza

6.

23a

41.

33s

1/4.64,

1700

and similarly,Rc(0.17)/R(10.6)= 1/15.7. The der-

ived radii are slightly above the radii calculated

(denotedby ●) in the optimized performancevs

wavelength simulationsfor Fig. 7 (b and c), which

provide a cross-checkon the procedure. The half-

intensitiesat 1.06 u and 0.17 p were obtained by

using the C02 peak powers and the derived critical

radii. Since Fig. 7(c) showed that the peak power

requirementsrise above the CO, values as we go to

shorter wavelength, the precise derived half-inten-

sities should be somewhat higher than indicated.

On Figure 8(d) we have included the initial

pellet edge radius Re(t=O) vs its mass. It is in- .

teresting to note that at pesk power the 1O.6-P

critical surface has moved out considerablyfrom

its initial position [i.e., from Rc(t=O)], the

1.06-u surface is essentiallyunmoved, and the

0.17-l.Icritical radius has descended to one-third

its initial value.

Examinationof the Table I(b) entries ahowa

that for lower peak intensities,~(tl)/47rR:, and

the resultant favorable reduction in hot electron

production, it is desirable to go to the most

massive pellet feasible under the energy constraints

IV. CONCLUSION

We have given detailed results from computer

calculationsof the hydrodynamicsand burn of op-

timally implodedDT spheres. We have shown how

conditionsin the pellet core prior to bum are

affected by variations in the pulse shape parameters.

Burn performancewas related to these pellet cond-

itionswith the aid of our earlier burn study re-

sults. For yields exceeding breakeven considerable

precision in the pulse shape is demanded. Degraded

performance is anticipatedfrom the presence of

hyperthermalelectrons generated at the high peak

laser power levels required for optimized sphere

implosions. Both the hyperthermalproductionand

the precizlionneeds can be reduced by going to

larger pellets and correspondinglyhigher input

energies.

REFERENCES

1.

2.

I 3“

S. Neddermeyer,aa described in D. Hawkins, LA=report No. LANS-2532, 1961 (unpublished),VO1.I, p. 23.

J. Nuckolls, T. Wood, A. Thiesaen, and G. Zim-merman, Nature (London)239, 139 (1972).—

J. S. Clarke, H. N. Fisher, and R. J. -son,Phys. Rev. Lett. 30, 89(1973),and Phys. Rev.Lett. ~, 249 (1973).

K. Brueckner, IEEE Transactionson PlasmaSciences PS-1, 13 (1973).

K. Boyer, Bull. Am. Phys. Sot. 1.7,1019 (1972),and Aeronautics and Astronautics=(l), 28(1973).

J. Nuckolls, J. Emmett and L. Wood, PhysicsToday, August (1973).

G. S. Fraley, E. J. Linnebur, R. J. Meson and

R. L. Morse, Phys. FluLda ~, 474(1974).

D. Forslund, J. Kindel and E. Lindman, Phys.Rev. Lett. 30, 739 (1973).—

of the available laser system.

TABLE I. Optimized pellet performancetiCS VS mSSS; P = 1.875.

characteris-

4.TABLE1 (a)

,

.

5.

6.

7.

8.

0.0.4 0.9 --

0.11 3.0 1.2

0.27 7.2 3.8

0.70 13.3 8.7

7.50 33.6 22.0

22.0 39.0 29.0

58.0 49.0 38.0

250. o 65.0 55.0

1.60X104 0.58 19.5 35

1.7X204 1.03 11.9 50

1.31X104 1.35 8.6 67

9.5X103 1.75 7.5 92

4.3X103 2.3s 6.6 203

3.OX1O3 2.65 6.9 291

2.1X103 2.94 7.3 .’402

1.O3X1O3 3.10 7.4 654

18

.

r

.

9.

10.

11.

12.

13.

14.

15.

16.

L. Spitzer, Physics of ~ Ionized Gases 17.(Interscience,New Y~k, 1969==

J. L. Tuck, Nucl. Fusion~, 202 (1961). 18.

H. Brysk, P. FL Csmbell, and P. Hammering,KMS Fusion Report No. V11O (1973). 19.

R. Courant and K. O. Friedrics,Pure and&——-~thematics VO1. I, SupersonicFlow and 20.——Shock Waves (Interscience,New York 1948),z =.

C. Fauquignonand F. Floux, Phys. Fluids 13,386 (1970).

—

J. L. Bobin, Phys. Fluids~, 2341 (1971).

R. L. Morse and C. W. Nielson, Phys. Fluidsl&, 909 (1973).

D.W. Forslund,J. Geophys. Res 7&, 17 (1970).

21.

T. M. O’Neil and M. N. Rosenbluth, IDA ReportNo. P-893 (1972).

R. E. Kidder and J. W. Zink, Nucl. Fus. 12,—325 (1972).

K. Lee, D. B. Henderson,W. P. Gula, and R. L.Morse, Bull. Am. Phys. SO.. ~. 1341 (1973).

J. P. Christiansen,and D. E. Ashby, SixthEuropean Conferenceon Controlled Fusion andPlasma Physics, Moscow, USSR. 30 July - 3August 1973.

P.A.G. Scheuer,Monthly Notices Roy. Astron.SOC. ~ 231 (1960).

APPENDIX

We calculate theinverse-bremsstrahlungenergy

absorption,using Scheuer’s21free-free coefficient

Ka=me6~3~2n2 -lf2

3 cs(2mekTe)3/2 (l-n<nc) (lnA’-5y)

(A-1)

Trmeczin which the critical density is .C = ~, A’

= (8k3Te3A2)/lr2mec2e4Z2), and y ia Euler’s constant,

0.577...

Electrons iU the computationalzones of width

k acquire specific energy via inverse-bremsstrah-

lung at a rate

i = (l-e-Kam) e-ZKaA%(t)/Am. (A-2a)‘i

Here the sum, XKaAR, is taken from the pellet

edge to the zone just before the critical density,

and fi(t)is the laser power (1.). The calculations

assume total anomalous absorption of all of the

remaining energy by dumping it into”the thermal

electrons in the firat cell where n exceeds nc.

Per unit time and mass this is

These two rates plus the specific burn energy rede-

position rate constitute the A, source term i. Eq.

(b-2a)of Ref. 7.

Tapered zoning was employed to improve the

resolutionand efficiencyof the calculations.

Thus, initially, the zoning was finer at the pellet

center and edge, than in its midregions. At the

center AR/Re(t=O) = 5 x 10-3. The neighboring

zones increase in size by the ratio AR~l/ARm=l.2.

At the edge AR/Re(t=O)= 1.3 x 10-3, while the

lower zones are larger by &m_lfARm= 1.053.

This generates 81 zones in total. For the 7.5+g

pellet, Re(t=O) = 203 p, the starting zone widths

were 1.01 p at the center, 0.28 p at the edge, and

the llth zone from the center was the largeat--

1.4 u.

ie = e-XKaAR(t)/Am.a

(A-2b)

EE:278(120)

19